New Guppy Momma, please correct me if I am mistaken, but lyretail mollies usually have extended rays on the outsides of the caudal or tail fin. The dorsal fins may be impressive too. However aren't the modified anal fins (or gonopodiums) the same as their parents among regular guppies? They should then be able to reproduce with females.

However Google Lyretail swordtails and look at the males. All fins are elongated. Females will have longer but less elongated fins. And it doesn't matter for them because those are not used for mating. That also happens with lyretail platys.

You are right on about not using the first fry dropped. And if the male was put into the aquarium the same day as the female when she dropped, if there was a chance that she could be made pregnant, that would be the time. In fact, since there may not room to keep and raise those fry, the male could be kept with her from even before the moment she drops.

If (and that is a pretty big IF) Josh's male was effective, some or all of the next batch could show the lyretail characteristics. If there were 30 or so fry, one would need at least a five or a 10-gallon tank to get them started. After a couple of months of militant weekly water changing ;) the larger dozen or so fry could be moved to a 20-gallon tank. Probably within three to seven months (space, temperature, feedings, water changes are among the variables) he should know if there were any lyretails.

If he wants to try that and understands that the odds are pretty slim that a male and female lyretail swordtail can produce fry (the Chicago Cubs probably have much better odds of winning the World Series), go for it. Just remember if that approach worked, the professional breeders and fish farmers would use the lyretail males in the breeding process. :)
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Here by the way is an article by a guy who was working to cross a high-fin swordtail with an albino strain. He details his two-year effort. Like New Guppy Momma, he used a 75-gallon tank to raise up that first batch of swords. The size of the aquarium (probably plus good maintenance and care) allowed him to raise them to adulthood in a a quick 3 to 4 months. At the time of the writing, when he was raising a fourth generation, he was still working on fixing all of the characteristics he sought.

http://www.aquarticles.com/articles/breeding/Lynch_Swordtail_Breeding_Program.html

That site offers a lot of great reading for aquarists. There are several useful guppy articles. The following is an article about the developing of what he called "the ultimate of black swordtails the Black Hi Fin Lyretail." He
too is very big on raising the swordtails in large tanks and very carefully selecting the breeding stock.